Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(4,5)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_2^2\times A_5^2$ |
| Order: | \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2\times A_5^2$ |
| Order: | \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $S_5\wr C_2$, of order \(28800\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5^{2} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient is nonabelian, an A-group, and nonsolvable.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5^2:D_6$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $-10800$ |
| Projective image | $C_2\times A_5^2$ |